To travel from town to town performing theatrical shows, or to campaign for political office, or to perform aerobatic flying displays.
From barn (meeting place) plus storm (to move forcefully), likely combining the idea of traveling to rural barns for performances with the energy of a storm passing through.
Early aviators literally barnstormed—they landed small planes in farmer's fields and performed aerial tricks, turning rural barns into impromptu airshows that amazed communities that had never seen flying!
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